Users' Mathboxes Mathbox for Stefan O'Rear < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  sbc4rex Structured version   Visualization version   GIF version

Theorem sbc4rex 42086
Description: Exchange a substitution with four existentials. (Contributed by Stefan O'Rear, 11-Oct-2014.) (Revised by NM, 24-Aug-2018.)
Assertion
Ref Expression
sbc4rex ([𝐴 / 𝑎]𝑏𝐵𝑐𝐶𝑑𝐷𝑒𝐸 𝜑 ↔ ∃𝑏𝐵𝑐𝐶𝑑𝐷𝑒𝐸 [𝐴 / 𝑎]𝜑)
Distinct variable groups:   𝐴,𝑏   𝐴,𝑐   𝐵,𝑎   𝐶,𝑎   𝑎,𝑏   𝑎,𝑐   𝐴,𝑑   𝐴,𝑒   𝐷,𝑎   𝐸,𝑎   𝑎,𝑑   𝑒,𝑎
Allowed substitution hints:   𝜑(𝑒,𝑎,𝑏,𝑐,𝑑)   𝐴(𝑎)   𝐵(𝑒,𝑏,𝑐,𝑑)   𝐶(𝑒,𝑏,𝑐,𝑑)   𝐷(𝑒,𝑏,𝑐,𝑑)   𝐸(𝑒,𝑏,𝑐,𝑑)

Proof of Theorem sbc4rex
StepHypRef Expression
1 sbc2rex 42084 . 2 ([𝐴 / 𝑎]𝑏𝐵𝑐𝐶𝑑𝐷𝑒𝐸 𝜑 ↔ ∃𝑏𝐵𝑐𝐶 [𝐴 / 𝑎]𝑑𝐷𝑒𝐸 𝜑)
2 sbc2rex 42084 . . 3 ([𝐴 / 𝑎]𝑑𝐷𝑒𝐸 𝜑 ↔ ∃𝑑𝐷𝑒𝐸 [𝐴 / 𝑎]𝜑)
322rexbii 3123 . 2 (∃𝑏𝐵𝑐𝐶 [𝐴 / 𝑎]𝑑𝐷𝑒𝐸 𝜑 ↔ ∃𝑏𝐵𝑐𝐶𝑑𝐷𝑒𝐸 [𝐴 / 𝑎]𝜑)
41, 3bitri 275 1 ([𝐴 / 𝑎]𝑏𝐵𝑐𝐶𝑑𝐷𝑒𝐸 𝜑 ↔ ∃𝑏𝐵𝑐𝐶𝑑𝐷𝑒𝐸 [𝐴 / 𝑎]𝜑)
Colors of variables: wff setvar class
Syntax hints:  wb 205  wrex 3064  [wsbc 3772
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2697
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-tru 1536  df-ex 1774  df-nf 1778  df-sb 2060  df-clab 2704  df-cleq 2718  df-clel 2804  df-nfc 2879  df-ral 3056  df-rex 3065  df-v 3470  df-sbc 3773
This theorem is referenced by:  6rexfrabdioph  42096  7rexfrabdioph  42097
  Copyright terms: Public domain W3C validator